class: center, middle, inverse, title-slide .title[ # Class 2a: Review of concepts in Probability and Statistics ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> ### Summary - In the last class: - We discussed the organization of the course - We overviewed forecasting methods - We learned about methods of qualitative forecasting - *Reference:* Forecasting Methods and Applications, chapter 1 - This set of classes: - We will start learning about exploratory analysis preparing the forecast - We will learn about various **data types** - We will learn how to **summarize data graphically** - We will learn how to **summarize data with summary statistics** - We will learn about **comparisons and associations** - *Reference:* Forecasting Methods and Applications, chapter 2.1-2.4 --- ### Scenario - Nowadays, many online pharmacies appeared which write prescriptions and make drugs subscriptions - Example in Mexico: *Choiz* -- - At the same time, a new wave of very effective anti-diabetes drugs appeared which help to lose weight - Example: *Ozempik* -- - You are consulting a business which wants to provide subscription services for these drugs in Mexico -- - Your boss asks you to do exploratory market research for potential sales forecast --- ### Parameters vs Statistics - You need to know how many people in Mexico have diabetes - Call `\(\mu_d\)` the proportion of Mexican population which has diabetes - This is a parameter that you want to learn - But you don't have data on the whole population. At best you can get a sample from a survey - So you will try to estimate this parameter with sample - You will calculate a statistic `\(\hat{\mu}_p\)` which is the proportion of diabetics in the sample --- layout: false class: inverse, middle # Types of Data --- ### Longitudinal Data - Observations are collected for the same subject over a period of time - Example: Tracking a company's annual revenue and number of employees over several years #### Longitudinal Data Example <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-f67acfb0ab41a4124c66" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-f67acfb0ab41a4124c66">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2018,2019,2020,2021,2022],[50000,52000,55000,58000,60000],[50,55,60,65,70]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Revenue<\/th>\n <th>Employees<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,2]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Cross-Sectional Data - Observations are collected at a single point in time - Example: A survey of customers' satisfaction with a product and likelihood of repurchase at a certain point in time #### Cross-Sectional Data Example <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-5d3784c68f6bead26537" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-5d3784c68f6bead26537">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4],[7,8,5,9],["Likely","Unlikely","Likely","Likely"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Customer_ID<\/th>\n <th>Satisfaction_Score<\/th>\n <th>Repurchase_Likelihood<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Panel Data - Combines both longitudinal and cross-sectional data - Observations are collected for multiple subjects over multiple points in time - Example: Tracking the annual revenue and number of employees of several companies over a few years #### Panel Data Example <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-19ebac2509a70925da32" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-19ebac2509a70925da32">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2018,2018,2018,2019,2019,2019,2020,2020,2020,2021,2021,2021,2022,2022,2022],["A","B","C","A","B","C","A","B","C","A","B","C","A","B","C"],[50000,52000,55000,58000,60000,60000,63000,65000,68000,70000,70000,72000,75000,78000,80000],[50,55,60,65,70,100,105,110,115,120,150,155,160,165,170]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Company<\/th>\n <th>Revenue<\/th>\n <th>Employees<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,2,3]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ## Q1 <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-9cddd6f0355441d6bf92" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-9cddd6f0355441d6bf92">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Jan","Jan","Jan","Feb","Feb","Feb","Mar","Mar","Mar","Apr","Apr","Apr","May","May","May","Jun","Jun","Jun","Jul","Jul","Jul","Aug","Aug","Aug","Sep","Sep","Sep","Oct","Oct","Oct","Nov","Nov","Nov","Dec","Dec","Dec"],["Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin"],[60000,40000,10000,62000,41000,10500,63000,42000,11000,64000,43000,11500,65000,44000,12000,66000,45000,12500,67000,46000,13000,68000,47000,13500,69000,48000,14000,70000,49000,14500,71000,50000,15000,72000,51000,15500]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Month<\/th>\n <th>Cryptocurrency<\/th>\n <th>Market_Cap<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Panel data** -Multiple observation per subject (currency) --- ## Q2 <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-92734c20253fa4f1c619" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-92734c20253fa4f1c619">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["United States","China","India","Brazil","Russia"],[331,1439,1380,213,145],[21433,15308,3160,1848,1690],[246,904,560,126,116]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Country<\/th>\n <th>Population_Millions<\/th>\n <th>GDP_Billions<\/th>\n <th>Internet_Users_Millions<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2,3]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Cross-sectional data** -Single observation per subject (user) --- ## Q3 <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-d0aee4241f7a2c27f260" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-d0aee4241f7a2c27f260">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2020,2021,2022,2023,2024],[20000,30000,40000,50000,60000]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Electric_Car_Sales<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Longitudinal data** -Multiple observations of a single subject --- layout: false class: inverse, middle ## Variable Types --- ## Variable Types We have two general types: .blue[Categorical] and .blue[Numerical] variables ### Categorical Variables - Variables that can be divided into one or more groups or categories. - **Ordinal:** These variables can be logically ordered or ranked. - *Variable:* Customer Satisfaction Survey Results - *Example:* Very Unsatisfied, Unsatisfied, Neutral, Satisfied, Very Satisfied - **Nominal:** These variables cannot be ordered or ranked. - *Variable:* Social Media Platforms Used - *Example:* Facebook, Instagram, Twitter, LinkedIn, TikTok, Snapchat --- ### Numerical Variables - Variables that hold numeric data. - **Discrete:** These variables can only take certain values - *Example*: Number of App Downloads from App Store - *Example*: Number of children you have - **Continuous:** These variables can take any value within a range - *Example*: Time (in seconds) Spent on a Webpage - *Example*: Exchange rate between MXN and USD --- ### Mexican Health Survey Representative sample of the Mexican population .red[n=37858] <div class="datatables html-widget 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class=\"display\">\n <thead>\n <tr>\n <th>weight<\/th>\n <th>n_i<\/th>\n <th>p_i<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] - Most values never repeat, so they have very low relative frequency --- ## Histograms **Solution**: Group similar values together - Construct intervals and show how many observations are in a given interval -- **Process** 1. Decide how many intervals 2. And how wide they are 3. Then calculate the absolute and relative frequencies of each interval 4. Plot it with bars -- --- **My approach** - I want `\(k\)` (example `\(k\)`=5) equal intervals -- - Divide the range of the data into `\(k\)` equal intervals -- - *Range* is max-min of the data -- ```r # Calculate max and min max_value <- max(Health_data$weight) min_value <- min(Health_data$weight) # Calculate the difference range <- max_value - min_value ``` -- ``` ## [1] "Range= 190.8078 - 30.3745 = 160.4333" ``` -- - With 5 intervals, each will be 32kg wide -- - The first one starts at the minimum value (30.3745) -- - The last one ends at the maximum value (190.8078) -- - Calculate how many observations I have in each interval and what's the relative frequency --- ## Histograms - Midpoint represents middle of the interval - center of the bar - `\(P_i\)` is cumulative frequency: share of observations in this or smaller interval - *Example*: `\(P_{(62.46-94.55)}=0.911\)` - *Interpretation*: 91.1% of people have weight lower than 94.55kg .pull-left[ <!-- --> ] .pull-right[ <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-bcb7704b8d49e60f6149" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-bcb7704b8d49e60f6149">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["30.37 - 62.46","62.46 - 94.55","94.55 - 126.63","126.63 - 158.72","158.72 - 190.81"],[46.42,78.5,110.59,142.68,174.76],[10068,24430,3206,143,11],[0.2659411,0.6453061,0.08468489999999999,0.0037773,0.0002906],[0.2659411,0.9112472,0.9959321,0.9997094,1]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Interval<\/th>\n <th>Midpoint<\/th>\n <th>n_i<\/th>\n <th>p_i<\/th>\n <th>P_i<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2,3,4]}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] --- ## Histogram with 10 Classes Now, let's increase the number of classes to 10. .pull-left[ <!-- --> ] .pull-right[ <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-f2c194872060ce579177" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-f2c194872060ce579177">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["30.37 - 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Helps to see the distribution and outliers - Is more always better? - With smaller intervals, histogram tends to the **probability density function** --- ## Probability Density Function (PDF) ### Definition - **Probability Density Function (pdf)** describes the probability distribution of a continuous random variable. - The PDF represents the likelihood of the random variable taking on a specific value within a given range. - Unlike the probability mass function (PMF) for discrete variables, the PDF does not give the probability of exact values but provides the relative likelihood of the random variable being within a certain interval. -- ### Example Suppose we have a random variable X representing the weight of adults in Mexican population. 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100, 100, 0.3)","marker":{"color":"rgba(255,127,14,1)","line":{"color":"rgba(255,127,14,1)"}},"error_y":{"color":"rgba(255,127,14,1)"},"error_x":{"color":"rgba(255,127,14,1)"},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## Distribution Shapes: Modality <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/unnamed-chunk-37-1.png" width="100%" /> --- ## Which is uniformaly distributed 1. weights of adult females 2. salaries of a random sample of people from CDMX 3. House prices in CDMX 4. birthdays of classmates (day of the month) --- ## Distribution Shapes: Modality <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/unnamed-chunk-38-1.png" width="100%" /> --- ### Age at death <center> <img src=Age_at_death.jpeg width="800"> </center> --- ## We want to know how many people weight more than 100kg --- ## Cumulative Distribution Function (CDF) The .blue[Cumulative Distribution Function] (CDF) gives the probability that a continuous random variable X will take on a value less than or equal to a specific value x. For a continuous random variable X with PDF f(x), the CDF F(x) is defined as: `\(F(x) = \int_{-\infty}^{x} f(t) \, dt = P(X<x)\)` The CDF provides a cumulative view of the probability distribution and is useful for calculating probabilities over intervals and finding percentiles. - The CDF starts at 0 as probability of X being less than or equal to negative infinity is 0 - It approaches 1 as x approaches infinity since the the probability of X being less than or equal to positive infinity is 1. --- ## Example 1: Standard Normal `\(F(-2) = \int_{-\infty}^{-2} f(t) \, dt = P(X<-2)=0.02\)` <!-- --> --- ## Example 2: Standard Normal `\(F(0.2) = \int_{-\infty}^{0.2} f(t) \, dt = P(X<0.2)=0.58\)` <!-- --> --- ## Example 3: Standard Normal `\(F(3.2) = \int_{-\infty}^{3.2} f(t) \, dt = P(X<3.2)=0.99\)` <!-- --> --- ### Empirical CDF We can do similar thing with our weight data. `\(ECDF(x)=\frac{\sum I(w_i<x)}{N}=\frac{\text{Number of people with weight lower than x}}{N}\)` - `\(I(w_i<x)=1\)` if weight of person `\(i\)` is lower than `\(x\)` (*Indicator Function*) - `\(N\)` is total number of people (*Sample Size*) - Share of people with weight lower than x <div id="htmlwidget-a804cd50813264ff574a" style="width:800px;height:300px;" class="plotly html-widget "></div> <script type="application/json" 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